Connecting geometry, nature and architecture
This is a new personal project that seeks to establish connections between various fields that interest me deeply: geometry, and how it appears linked to nature, on the one hand, and to art and architecture, on the other; raising relations between all of them. Somehow it would mean the fusion of the interests of two of my previous works: Nature by Numbers and Ars Qubica.
[ You can also watch this video on YouTube ]
And on the most technical side of things: the next video shows a bunch of screen-captures with the process of creating this animation, created side by side with Modo and Cinema4D.
The concepts behind Infinite Patterns
This section intends to be a complement to the animation, to better understand the theoretical base that is enclosed behind it. It was also, in part, the aspect of the script that I wrote when I was planning the project.
Zero, one and two dimensions
The animation starts by presenting three geometric figures, the three simplest regular polygons: an equilateral triangle, followed by a square and finally a hexagon. But it makes it a very special way. Let’s look at it in detail:
To begin with, the first thing that appears on the screen is a point, one of the “fundamental entities of geometry”, along with the line and the plane. A geometric figure without dimension, length, area or volume. The minimum unit of visual communication.
That point becomes a straight line, our second fundamental entity, with a single dimension and an infinite number of points.
And finally, on that line we define a segment that, by means of two turns centred in its extremes, allows us to obtain an equilateral triangle; the simplest regular polygon, with which (as with any triangle) we have already defined our third fundamental entity: the plane, with two dimensions. Notice: in these first seconds we have gone from less to more: rising from 0 to 1 and then to 2 dimensions ;-)
The particular way of transforming the equilateral triangle into a square and then a hexagon is not a whim. I needed to make it clear that these three particular regular polygons have the same area. Stay with this detail: all three are polygons of equal surface.
One way to visually represent the equality of areas between the three polygons is to cut them by a few lines and recompose the pieces to convert one into another. And to make it more elegant, do it in such a way that all parts are always in contact. It’s what’s known as “hinged dissection” or “Dudeney’s dissection.” A geometric dissection in which all the pieces are joined in a chain by “articulated” points, so that the conversion of one figure into another can be carried out by turning the chain continuously, without cutting any of the connections and without leaving small holes between the parts.
Next animated gif is from Wikipedia (Author: Rodrigo Silveira Camargo) and was of great help to understand and design this section of animation:
The transformation between an equilateral triangle and a square is relatively well known, as it has been widely used as a mathematical puzzle in many hobby books since mathematician Henry Dudeney popularised it in the early 20th century.
Curiously enough, the Wallace-Bolyai-Gerwien theorem, first tested in 1807, states that any two polygons of the same area will always have a common dissection. However, the question of whether two of these polygons should also share a hinged dissection remained open until relatively recently, in 2007, when Erik Demaine and other colleagues demonstrated that such a hinged dissection must always exist, and provided a constructive algorithm to produce them. Here is the PDF where that demonstration appears.
Hexagonal tiling is special
Good. But why is it so important, in this animation, to make it visually clear that the three polygons have the same surface? Because immediately after we will see how of these three figures, when unfolding in straight line each one of its contours, the one that has a smaller perimeter is the hexagon.
In the following diagram, which appears in the animation, we see what the difference is. We start from the premise that, by design, the area is identical in all three figures. Let’s say that it is exactly 1. It doesn’t matter if 1 square cm, 1 m or 1 square km…
To calculate the perimeter of the equilateral triangle we can use Heron’s formula which allows us to obtain the area of any triangle by knowing its sides. Here we do not know the sides, but we do know the area, so they are simple mathematics. There is also a formula for calculating the area of a hexagon by knowing its side. Applying it inversely, since we know the area, we can calculate the final perimeter.
And what does this imply?
There are infinite regular polygons (polygons whose sides and interior angles are equal), starting with the equilateral triangle, with only three sides, following with the square, pentagon, hexagon, heptagon, octagon… until ending with a circumference, which we could define as a regular polygon of infinite sides.
But of all the regular polygons there are only three with which you can tessellate the plane using the same piece: the equilateral triangle, the square and the hexagon (if we use combinations of two or more regular polygons we have many more possibilities).
And of those three alternatives, the hexagonal is the most compact. For example: if we were to divide fields into equal cultivated plots separated by stone walls, the optimal way (to spend the minimum number of stones) would be hexagonal tessellation. If not used in real life is probably because the square/rectangular is much easier to trace, even if you spend a few more stones ;-)
This is also known as the “honeycomb conjecture” (PDF) which states that this tessellation is the best way to divide a surface into regions of equal area and with the minimum total perimeter. A conjecture that has been known since antiquity, but was not proven until the end of the 20th century, in 1999 (!)
Finally, the third dimension: a honeycomb. And a bee
At this point, that’s what we see next in our animation: how tessellation hexagons are extruded to build a honeycomb. Also, look, we’ve already reached the third dimension, so our camera rotates to see everything in perspective.
The bees, which do not have a hair of fools, know that with this type of packaging they manage to create a structure with the optimal relation between the volume dedicated to the cavities and the wax necessary to build their walls. We will not go into more detail here, but the exact shape of the cells of a honeycomb raises other geometric curiosities.
Once our honeycomb has been built, a worker bee appears, who realizes that there is an empty cell, without honey, and decides to go out to work to solve it ;-)
And in the meantime, we move on to the next section of animation.
Islamic decorative patterns. Alhambra
The three-dimensional honeycomb is simplified until it again reaches a flat hexagonal tessellation. And from there, through a succession of geometric operations, we will obtain a much more complex decorative tessellation. In the following animated gif you can see in detail the succession of steps to get there.
I recommend that you take a look at the Tilingsearch.org website, which contains a multitude of very detailed diagrams (including PDFs) showing the construction of many other tessellations typical of Islamic art. Here is the review of the one we used in our animation, which is in the Hall of the Kings of the Alhambra in Granada.
And here, at left, you can see the page of a book that served as the basis for its geometric construction. The source for this image is another interesting site devoted to decorative tilings: PatternsInIslamicArt.com (Catalog No PIA 060)
And of course, it is to the Hall of the Kings where we arrive in the next shot and from there we go out to the famous Court of the Lions.
Since many years ago I visited Granada for the first time and discovered its greatest jewel, the Alhambra, I had always wanted to include it in some of my personal projects. This spectacular group of palaces and gardens enclosed in a fortress, which in times became a city within Granada itself and which was built over several centuries, has always seemed to me a beautiful work, almost overwhelming. I can’t get enough of visiting this gem, over and over again.
And this seemed to me to be the perfect occasion, because it is not only the masterpiece of Al-Andalus art, it is also one of the best places in the world to enjoy Islamic decoration through geometric patterns in its mosaics, tiles and stuccoes. Not in vain is the second tourist attraction in Spain, only surpassed by Gaudí’s Sagrada Familia.
As the camera moves through the Court of the Lions, I have allowed myself the license to include an unlikely element (a small artistic license, if you prefer): a flower that appears from the slits formed by the white marble slabs, right next to one of the water gutters.
Here comes again the golden angle
As we get closer, the flower grows and opens, to discover that it is a daisy. And the central disc of this flower, presents a very characteristic yellow structure, an inflorescence with a distribution that follows the same geometric pattern as the seeds of the sunflower, based on the golden angle (137.507º). In the section “The concepts behind…” of my work Nature by Numbers you will find details on how this interesting structure is formed.
And here we have again our worker bee that comes to look for her raw material on the daisy, in order to be able to fill the empty cell of her honeycomb.
The eyes of the bees
But on this second occasion the camera comes very close to one of the bee’s eyes (yes, it is also a wink at the end of Nature by Numbers) to discover that they are composed of multiple structures, the ommatidium distributed according to a hexagonal pattern (picture with compound eye at right created by Kils, from Wikimedia)
As a curiosity, each ommatidium projects its own image. The drone bees have about 8000 ommatidium (because they must have good eyesight to be able to locate the queen), the workers about 5000 and the queen about 4000 (because she will only come out four or six times from the nest in her lifetime).
In addition, bees also have three small simple eyes (ocelli) on the top of their heads, which are used for distant vision and to better perceive the intensity of light. By the way: bees distinguish blue, yellow and white but cannot see red.
From atoms to DNA. The basis of life
Once this hexagonal surface (the enormously enlarged eye of the bee) fills our screen, it serves as a support to generate the structural formulas of four molecules that are fundamental for life: the nitrogenous bases Adenine, Guanine, Cytosine and Thymine, which are usually simplified to their initials A-G-C-T. Next graphics are from Wikipedia, author Vesprcom:
All of them have a hexagon in their structural formula. And they all share and are linked by atoms:
- Carbon, in dark grey and no letter in the animation (as this is usually done in chemical notation)
- Nitrogen, in blue and with the letter (N)
- Hydrogen, blank and with the letter (H)
- In addition, Guanine, Cytosine & Thymine also contain Oxygen atoms, in red and with the letter (O)
In the animation you can see how, once formed, these four nucleobases begin to link. But they do not do it in an arbitrary way: Adenine (A) always connects with Thymine (T) and on the other hand Guanine (G) always connects with Cytosine (C). So the links are always either AT/TA or GC/CG.
And this succession of links between pairs, to which other molecules such as sugars and phosphates are also added, is what ends up forming the very famous structure of DNA, with its characteristic double helix shape, which contains the genetic instructions used in the development and functioning of all living beings and is in charge of hereditary transmission.
Double helix: from DNA to Bramante Staircase
The animation advances. With the large DNA molecule already perfectly formed, several helical lines are drawn around it. These lines of light expand and from them a somewhat mysterious construction begins to develop. Several steps and complex reliefs with ornamental motifs (angels, eagles and vegetable ornaments) all with a metallic and reflective appearance originate and connect on it.
Finally, from a zenithal point of view we see how the metallic finish is nuanced and we end up appreciating the structure of one of the most photographed architectural works by tourists in Rome: the modern Bramante Staircase in the Vatican Museums, with its richly ornamented balustrade in bronze. One of the few double helix staircases in the world, giving rise to perspectives with very suggestive curved lines.
Although it is not the original building piece: it is a relatively modern design, from 1932, made by Giuseppe Momo and inspired by the one designed by Donato Bramante in 1512, so that Pope Julius II could enter his private residence without leaving his carriage. In this other modern version, the initial purpose of the double helix was to make it possible for visitors to climb up and down without crossing each other. Although at present I’m afraid that it is only used in one direction, so part of the original intention has been lost…
Double helix: from Bramante Staircase to palm tree
Once we’ve walked through this double helix ladder-ramp, the camera rotates around the vertical pedestal at its base, with a large marble cup on top. Several beams of light emerge from it and a sort of circular grid is formed, reminiscent of a radar. A graph on which several spherical “particles” begin to appear, one after the other.
With each turn of 137.5º (again the golden angle) a new particle appears, but this time, unlike what happened in the sunflower of Nature by Numbers or in the daisy of the Alhambra, these particles are compacted to create a cylindrical shape, not a disc. The particles are transformed into a kind of flattened scales and the cylinder grows in height.
At a given moment, several helical curves emerge from the balustrade of the staircase that climb towards the centre while the architecture itself crumbles and disappears. And they suggest how, when adapting over the growing central cylinder, the scales of its surface (shaped, let us remember, by means of a golden distribution) also present a helical spiral arrangement. Really with spirals in two senses, although that detail doesn’t appear in the animation: too many things to show in such a short time!
This large cylinder that has grown in the center of our space is effectively the trunk of a palm tree. And finally we see how their enormous leaves appear in that distribution so characteristic of these great plants.
In fact, the “scales” that we perceive on the surface of the trunk of a palm tree are nothing other than its old “branches” (really leaves) that have already been falling or have been cut.
Speaking of palms: King’s College Chapel has the largest palmed vault
Then the big leaves of our palm tree fold again and the lines of a surface very similar to a truncated hyperboloid are generated. In fact it is not, strictly speaking, a hyperboloid, since this figure is a surface of revolution generated by the rotation of a hyperbola, which is not exactly the case here, as we shall see.
What we have here is the constructive basis of a great new architectural work: the King’s College Chapel in Cambridge. This fantastic building, finished at the beginning of the 16th century, has the largest fan vault in the world, also called the palmed vault, built between 1512 and 1515 by the master bricklayer John Wastell. Its impressive stained glass windows (12 on each side and two even larger at both ends) were mostly made by Flemish craftsmen between 1515 and 1531.
Four-centred arches an also ogival-golden ones. And a rose
One of the most characteristic elements of this architectural style is its four-centred arch (known precisely as the “Tudor Arch” outside England).
In the following animated gif we see how the arches for the great west gate are constructed using this type of four-point arches. In my preliminary research I also arrived at a happy discovery: as you can see, to obtain this diagram we started… from a hexagon! There are different geometric procedures for finding the centers for a four-point arc (depending on their final proportions). I cannot state categorically that this method shown here is the one actually used by original builders, but it certainly fits the references very well:
For the twelve side windows, instead, pointed arches (also called ogival) were used, made up of two sections of circumferential arches that form an angle in the keystone. And in this case I found it highly probable that for its construction it was based on a golden rectangle, as you can see in the following gif:
After a short tour of the inside of the chapel, we approached the top of the vault with one of the ornamental elements that is repeated throughout the interior: a large stone rose (symbol of the two great dynasties that intervened in the War of Roses, the House of Lancaster, with a red rose, and the House of York, with a white rose).
At the last moment that rose opens, let’s show a geometric structure in its central button, something begins to emerge and then… Fade to black. End.
A mystery: catenary curves? Really? Where?
Finally, I want to make a confession: when I was planning this project, my idea was to include a building that made use of catenary curves. The most logical or obvious choice could have been to use one of Gaudí’s many works, as he was one of the greatest exponents of the use of this type of curves in architecture. But… let’s say that Gaudí’s card is reserved “for another occasion” ;-)
So looking for other examples on the net I found several references to this other work, the King’s College Chapel, as a sample of the use of this type of curves, the catenaries. For example:
- In the Wikipedia article on the use of catenary in architecture (see first example of “Cathedrals and Churches”),
- On page 17 of the book “Observations on the Construction of the Roof of King’s College Chapel, Cambridge”
- In this video “The Catenary – Mathematics All Around Us” from minute 3:10
So, great, I found it! I loved the building and it was going to be a wonderful example of the use of catenary in architecture, perfect to close the animation as I had planned!
But when the moment of truth arrived, after downloading abundant documentation, photos, drawings, scanners of several books, etc. the first thing I did was to look exactly where those kind of curves appeared in this building. It is supposed to be in the vault, as catenaries are characterised precisely by generating a type of extremely resistant arches.
I even printed several elevations, from different angles, inverted them and used a little chain to try to find where the hell the catenaries were.
But no matter how much I looked, I couldn’t find a single curve or arch that reminded me of a catenary, anywhere (!?)
All this reminded me of the history of the golden spiral as the basis for the development of the shell of a nautilus, an anecdote that I explain in the section “The concepts behind…” of my old work Nature by Numbers. On that occasion, despite verifying that the spiral of a nautilus is NOT really a golden spiral, I decided to leave it as it is (as it is stated, erroneously, in countless articles and reports throughout the Internet) and excuse myself under the pretext of the “artistic license”.
But here at this new job, I didn’t want the same thing to happen again. Enough of forced “artistic licenses”…
So, continuing with my research, I came into contact with an architectural firm of which I had found a PDF publication describing a visit their workers had made to England to study this type of work (King’s College Chapel and others). After a first self-introduction, I sent them several documents and graphics and asked them all my questions and doubts. Basically, where were those catenary curves found specifically in the elevations or sections of this building? Adding this kind of attached graphic:
I must say I’ve been treated wonderfully by those professionals. And they finally recognised that, “perhaps”, the catenary had not been used to design the arches of the vault of this work, contrary to what is stated in several places…
So, I finally desisted from introducing the subject of catenary. In spite of everything, I’ll always keep the doubt. If any expert reads this and can confirm me in one way or the other, I’ll be grateful. But it’s not enough for me a simple statement like “of course the catenaries were used, I know it”. I need to see some clever diagram, something that original builders could plausibly use with the techniques and knowledge of that 15th century. And to say that “a portion of the arches for the vaults could correspond to a portion of a catenary curve…” that does not seem to me to be a serious argument.
Anyway, I finally found that the section for the main vault, while not responding to a design based on catenary curves, did so to one based on the arches of four centres. And besides, oh, surprise, it fit perfectly with a hexagon-based layout! Which was in fact a better fit with the rest of the short film. And on the other hand, the pointed arches for the windows matched a diagram based on the golden rectangle! I shouted “eureka” when I found these two coincidences, after my research ;-)
So I finally didn’t have to give up the inclusion of this singular building in my project, even though I didn’t found the catenary curves in its construction.
Cristóbal Vila, September 2019, Zaragoza, Spain